Kuhn Length Research Articles

Supersoft and Hyperelastic Polymer Networks with Brushlike Strands

Using a combination of the scaling analysis and molecular dynamics simulations, we study relationship between mechanical properties of networks of graft polymers and their molecular architecture. The elastic response of such networks can be described by replacing the brushlike strands with wormlike strands characterized by the effective Kuhn length which is controlled by the degree of polymerization of the side chains nsc and their grafting density 1/ng. In the framework of this approach we have established relationships between the network structural shear modulus G, strands extension ratio β, and architectural triplet [nsc, ng, nx], where nx is the degree of polymerization of the backbone strand between cross-links. Analysis of the simulation data shows that G could increase with β (G ∝ β), which reflects the “golden rule” of elastomers: softer materials are more deformable. However, networks of graft polymers can also break this rule and demonstrate an increase of the modulus G with decreasing extensio...

Unravelling cationic cellulose nanofibril hydrogel structure: NMR spectroscopy and small angle neutron scattering analyses.

Stiff, elastic, viscous shear thinning aqueous gels are formed upon dispersion of low weight percent concentrations of cationically modified cellulose nanofibrils (CCNF) in water. CCNF hydrogels produced from cellulose modified with glycidyltrimethylammonium chloride, with degree of substitution (DS) in the range 10.6(3)-23.0(9)%, were characterised using NMR spectroscopy, rheology and small angle neutron scattering (SANS) to probe the fundamental form and dimensions of the CCNF and to reveal interfibrillar interactions leading to gelation. As DS increased CCNF became more rigid as evidenced by longer Kuhn lengths, 18-30 nm, derived from fitting of SANS data to an elliptical cross-section, cylinder model. Furthermore, apparent changes in CCNF cross-section dimensions suggested an "unravelling" of initially twisted fibrils into more flattened ribbon-like forms. Increases in elastic modulus (7.9-62.5 Pa) were detected with increased DS and 1H solution-state NMR T1 relaxation times of the introduced surface -N+(CH3)3 groups were found to be longer in hydrogels with lower DS, reflecting the greater flexibility of the low DS CCNF. This is the first time that such correlation between DS and fibrillar form and stiffness has been reported for these potentially useful rheology modifiers derived from renewable cellulose.

Dynamic Looping of a Free-Draining Polymer

We revisit the celebrated Wilemski-Fixman (WF) treatment for the looping time of a free-draining polymer. The WF theory introduces a sink term into the Fokker-Planck equation for the $3(N+1)$-dimensional Ornstein-Uhlenbeck process of the polymer dynamics, which accounts for the appropriate boundary condition due to the formation of a loop. The assumption for WF theory is considerably relaxed. A perturbation method approach is developed that justifies and generalizes the previous results using either a Delta sink or a Heaviside sink. For both types of sinks, we show that under the condition of a small dimensionless $\epsilon$, the ratio of capture radius to the Kuhn length, we are able to systematically produce all known analytical and asymptotic results obtained by other methods. This includes most notably the transition regime between the $N^2$ scaling of Doi, and $N\sqrt{N}/\epsilon$ scaling of Szabo, Schulten, and Schulten. The mathematical issue at play is the non-uniform convergence of $\epsilon\to 0$ and $N\to\infty$, the latter being an inherent part of the theory of a Gaussian polymer. Our analysis yields a novel term in the analytical expression for the looping time with small $\epsilon$, which is previously unknown. Monte Carlo numerical simulations corroborate the analytical findings. The systematic method developed here can be applied to other systems modeled by multi-dimensional Smoluchowski equations.

The effect of RNA stiffness on the self-assembly of virus particles

Under many in vitro conditions, some small viruses spontaneously encapsidate a single stranded (ss) RNA into a protein shell called the capsid. While viral RNAs are found to be compact and highly branched because of long distance base-pairing between nucleotides, recent experiments reveal that in a head-to-head competition between an ssRNA with no secondary or higher order structure and a viral RNA, the capsid proteins preferentially encapsulate the linear polymer! In this paper, we study the impact of genome stiffness on the encapsidation free energy of the complex of RNA and capsid proteins. We show that an increase in effective chain stiffness because of base-pairing could be the reason why under certain conditions linear chains have an advantage over branched chains when it comes to encapsidation efficiency. While branching makes the genome more compact, RNA base-pairing increases the effective Kuhn length of the RNA molecule, which could result in an increase of the free energy of RNA confinement, that is, the work required to encapsidate RNA, and thus less efficient packaging.

Hierarchical structure of MWCNT reinforced semicrystalline HDPE composites: A contrast matching study by neutron and X-ray scattering

Combining neutron and X-ray scattering (SANS, SAXS, and WAXD) as the contrast matching probes, the hierarchical structures of the multi-wall carbon nanotubes/high density polyethylene (MWCNTs/HDPE) composites (0.1–2.0 wt%) have been studied. SANS measurement avoided the correlation peak associated to the semicrystalline nature of HDPE from SAXS, and the passible scattering from the stretch induced voids. Benefitting from SANS data, the power law (α) taking the flexible cylinder suggested that whole MWCNTs in the unstretched composites follow a self-avoiding random walk (α ≈ 1.67). SANS data suggest a hollow flexible cylinder structure with outer and inner diameters (core-shell-like cross section) of the scatterers as ca. 12.6 nm and 4.1 nm, while the Kuhn length is about 43.0 nm. Tensile test displayed improved mechanical properties. Interestingly, the Hermans orientation parameter (f) of HDPE crystals (fc) in the stretched composites are around a plateau ca. 0.96 initially, while decreased remarkably to 0.776 when the fraction is further increased to 2.0 wt%. f obtained by SANS showed a similar tendency (decreased from 0.756 to 0.572). Combining all the results along with differential scanning calorimetry (DSC) measurement, a mechanism of adhered crystals on the surface of MWCNTs and a reinforcement model are proposed.

Influence of Solvent Quality on the Force Response of Individual Poly(styrene) Polymer Chains.

Single molecule mechanics of poly(styrene) polymer chains is investigated in different organic solvents with atomic force microscopy (AFM). The acquired force-extension profiles can be well fitted with a modified freely jointed chain (FJC) model. The model describes the force-extension profiles in terms of an apparent Kuhn length and an elasticity constant. The elasticity constant is found to be the same for all different solvents investigated. Best fit of the force-extension profiles with the FJC model reveals that the Kuhn length varies systematically with solvent quality. In fact, one can establish a good correlation between the Kuhn length and the Flory-Huggins interaction parameter. The increase in the Kuhn length with increasing solvent quality reflects the larger extent of swelling of the polymer in good solvents.

Flow-Induced Nematic Interaction and Friction Reduction Successfully Describe PS Melt and Solution Data in Extension Startup and Relaxation

The uniaxial extensional-flow data of Huang and co-workers [ACS Macro Lett. 2013, 2, 741; J. Rheol. 2016, 60, 465] on several polystyrene systems, both melts and solutions, are here successfully compared with the predictions of a recent model [Ianniruberto, Macromolecules 2015, 48, 6306] that accounts for flow-induced friction-coefficient reduction. The model is here effectively simplified without loss of accuracy. Comparison with the solution data of Huang et al. [ACS Macro Lett. 2013, 2, 741], only differing for the size of the oligomeric solvent (from nearly one Kuhn length up to a few ones), reveals that friction reduction must include oligomer-size-dependent orientational coupling between the oligomeric solvent and the polymer molecules, such interactions vanishing for the smallest oligomer. The rheological “measure” of this nematic-like interaction as a function of the oligomer size (for small oligomers) is the novel result of the present paper, together with a further confirmation of the importance of friction coefficient reduction in strongly aligning flows. The model is also used to describe the stress relaxation data following fast uniaxial extension recently reported by Huang and Rasmussen [J. Rheol. 2016, 60, 465], confirming that the short-time relaxation process occurs at a faster rate because of the friction coefficient reduction induced by the previous flow. Those data also reveal the key role of convective constraint release induced by chain retraction, similarly to what happens in relaxation experiments following a large step strain.

Aberration-Corrected Electron Beam Lithography at the One Nanometer Length Scale.

Patterning materials efficiently at the smallest length scales is a longstanding challenge in nanotechnology. Electron-beam lithography (EBL) is the primary method for patterning arbitrary features, but EBL has not reliably provided sub-4 nm patterns. The few competing techniques that have achieved this resolution are orders of magnitude slower than EBL. In this work, we employed an aberration-corrected scanning transmission electron microscope for lithography to achieve unprecedented resolution. Here we show aberration-corrected EBL at the one nanometer length scale using poly(methyl methacrylate) (PMMA) and have produced both the smallest isolated feature in any conventional resist (1.7 ± 0.5 nm) and the highest density patterns in PMMA (10.7 nm pitch for negative-tone and 17.5 nm pitch for positive-tone PMMA). We also demonstrate pattern transfer from the resist to semiconductor and metallic materials at the sub-5 nm scale. These results indicate that polymer-based nanofabrication can achieve feature sizes comparable to the Kuhn length of PMMA and ten times smaller than its radius of gyration. Use of aberration-corrected EBL will increase the resolution, speed, and complexity in nanomaterial fabrication.

Light-responsive expansion-contraction of spherical nanoparticle grafted with azopolymers.

Due to the very importance for both fundamental research and technological applications, smart materials with stimuli-responsive properties have been studied intensively. Theoretical investigation contributes to this endeavor through constructing and analyzing a model system which captures main features of the corresponding complex material, wherefrom useful insight can be provided to the trial-and-error experiments. We here report a theoretical study on the smart spherical nanoparticle grafted with light-responsive azobenzene-containing polymers. Utilizing the photoisomerization ability of the azobenzene group, nanoparticles can undergo a light-induced expansion-contraction transition. The wormlike chain based single chain in mean field theory, which has been developed by us recently, is used to investigate this transition in detail. Exploring a large parameter space, our results definitely determine the parameters, including the chain length and effective Kuhn length of grafted chain, nanoparticle radius, grafting density, and position of the azobenzene group along the chain contour, to admit optimum light-responsive behavior of the smart nanoparticle, which provides a guide for experimentalists to design this type of material in a rational manner.

Conformational and electronic properties of small benzothiadiazole-cored oligomers with aryl flanking units: Thiophene versus Furan

Symmetrical benzothiadiazole-cored (BTZ) oligomers with aromatic flanks are widely used in experiments as structural blocks for organic electronic materials. Along with chemical composition, the molecular conformation plays a crucial role in crystal packing/self-assembly of these building blocks in thin films. In this study, we perform an extensive theoretical comparison of conformational preferences, electronic and optical properties of small π-conjugated molecules having BTZ central unit symmetrically decorated with thiophene (Th) or furan (Fu) rings using DFT calculations. In addition to the conformational screening of small molecules, the torsion potentials of the internal rotation and the energetics of weak intramolecular S⋯N, O⋯N, O⋯H and N⋯H nonbonded interactions stabilizing certain conformations are evaluated. The conformational properties of –[BTZ-Fu]n– and –[BTZ-Th]n– chains are predicted applying the hindered rotation model. Our calculation shows that substitution of one atom, here sulphur by oxygen, leads to huge stiffening of the resulting polymer as estimated by the Kuhn length based on the rotational isomeric model. On the other hand, the calculation of the band gaps and the UV–vis spectra shows that the electronic and optical properties of both compounds are almost identical. This offers the possibility to decouple the intramolecular electronic properties from the intermolecular arrangement and the morphology of the materials since the latter properties are sensitive to the local stiffness of oligomers and polymers.

Dynamics of Bottlebrush Networks

The deformation dynamics of bottlebrush networks in a melt state is studied using a combination of theoretical, computational, and experimental techniques. Three main molecular relaxation processes are identified in these systems: (i) relaxation of the side chains, (ii) relaxation of the bottlebrush backbones on length scales shorter than the bottlebrush Kuhn length (bK), and (iii) relaxation of the bottlebrush network strands between cross-links. The relaxation of side chains having a degree of polymerization (DP), nsc, dominates the network dynamics on the time scales τ0 τsc, bottlebrush elastomers behave as networks of filaments with a shear modulus G0BB(t) ∼ (nsc + 1)−1/4t–1/2. Finally, the response of the bottlebrush networks becomes time independent at...

A Simple Analytical Model for Predicting the Collapsed State of Self-Attractive Semiflexible Polymers.

We develop an analytical model to predict the collapse conformation for a single semiflexible polymer chain in solution, given its length, diameter, stiffness, and self-attractiveness. We construct conformational phase diagrams containing three collapsed states, namely torus, bundle, and globule over a range of dimensionless ratios of the three energy parameters, namely solvent-water surface energy (), energy of bundle end folds (), and bending energy per unit length in a torus (). Our phase diagram captures the general phase behavior of a single long chain (>10 Kuhn lengths) at moderately high (order unity) dimensionless temperature, which is the ratio of thermal energy to the attractive interaction between neighboring monomers. We find that the phase behavior approaches an asymptotic limit when the dimensionless chain length to diameter ratio (L*) exceeds 300. We successfully validate our analytical results with Brownian Dynamics (BD) simulations, using a mapping of the simulation parameters to those used in the phase diagram. We evaluate the effect of three different bending potentials in the range of moderately high dimensionless temperature, a regime not been previously explored by simulations, and find qualitative agreement between the model and simulation results. We, thus, demonstrate that a rather simplified analytical model can be used to qualitatively predict the final collapsed state of a given polymer chain.

Impact of Conformational and Chemical Correlations on Microphase Segregation in Random Copolymers

Random copolymers play an important role in a range of soft materials applications and biological phenomena. An individual monomer is typically a single chemical unit whose length is comparable to or less than a Kuhn length, resulting in a monomer segment that is structurally rigid at length scales of a segregated domain. Previous work on random copolymer phase segregation addresses the impact of correlations between the chemical identities along the chains for flexible polymers. In these works, a single monomer unit is effectively a large polymer block that behaves as a random walk without conformational correlation associated with semiflexibility. In our work, we develop a model of semiflexible random copolymers using the wormlike chain model to capture conformational correlation of the polymer chains. To address the thermodynamics of microphase segregation and the structure of the segregated domains, we develop a random phase approximation up to quartic order in density fluctuations that leverages our ...

Interaction of Polymers with Single-Wall Carbon Nanotubes

Polymers are widely used for postsynthesis processing, purification, and individualization of single-wall carbon nanotubes (SWNTs) in aqueous or organic solvent environments. Here, the interaction of single-stranded DNA oligomers (ssDNA) and of a polyfluorene copolymer (F8T2) with (6,5) SWNTs was investigated from desorption kinetics and in the case of ssDNA also using adsorption isotherms. Eyring analysis of desorption rate constants reveals a linear increase of activation enthalpies with ssDNA oligomer length until ΔdesH‡ saturates at (155 ± 5) kJ·mol–1 for oligomers exceeding the ssDNA Kuhn length of about 6 nm. The Gibbs energy for desorption of ΔdesG‡ = (96 ± 1) kJ·mol–1 is length-independent because of entropy–enthalpy compensation. The saturation of desorption energies at the high polymer coverages studied here is attributed to incomplete adsorption with typically no more than a single Kuhn segment of a polymer attached to a SWNT.

Network dynamics in nanofilled polymers.

It is well accepted that adding nanoparticles (NPs) to polymer melts can result in significant property improvements. Here we focus on the causes of mechanical reinforcement and present rheological measurements on favourably interacting mixtures of spherical silica NPs and poly(2-vinylpyridine), complemented by several dynamic and structural probes. While the system dynamics are polymer-like with increased friction for low silica loadings, they turn network-like when the mean face-to-face separation between NPs becomes smaller than the entanglement tube diameter. Gel-like dynamics with a Williams–Landel–Ferry temperature dependence then result. This dependence turns particle dominated, that is, Arrhenius-like, when the silica loading increases to ∼31 vol%, namely, when the average nearest distance between NP faces becomes comparable to the polymer's Kuhn length. Our results demonstrate that the flow properties of nanocomposites are complex and can be tuned via changes in filler loading, that is, the character of polymer bridges which ‘tie' NPs together into a network.

Illumination of Conjugated Polymer in Solution Alters Its Conformation and Thermodynamics

The importance of chain structure in conjugated polymer-based material active layers and its relation to device efficiencies in OPVs, organic field transistors, OLEDs, and other devices has been well established. However, the influence that the absorbance of the light inherent to these devices might have on the conjugated polymer structure is not well understood. Herein, we employ small-angle neutron scattering to investigate structural changes occurring in solutions of poly(3-hexylthiophene-2,5-diyl) with exposure to white light. Results indicate significant decrease in both Kuhn length (b) and radius of gyration (Rg) of the polymer upon illumination, coupled with a drop in the second virial coefficient (A2). We explain this phenomenon through a chain collapse model, proposing that the interaction of light with the polymer backbone alters its thermodynamic interactions with and solubility in the surrounding solvent. The presence of such an effect, which we observe in several conjugated polymers, introduc...

Anisotropy of the monomer random walk in a polymer melt: local-order and connectivity effects

The random walk of a bonded monomer in a polymer melt is anisotropic due to local order and bond connectivity. We investigate both effects by molecular-dynamics simulations on melts of fully-flexible linear chains ranging from dimers (M = 2) up to entangled polymers (M = 200). The corresponding atomic liquid is also considered a reference system. To disentangle the influence of the local geometry and the bond arrangements, and to reveal their interplay, we define suitable measures of the anisotropy emphasising either the former or the latter aspect. Connectivity anisotropy, as measured by the correlation between the initial bond orientation and the direction of the subsequent monomer displacement, shows a slight enhancement due to the local order at times shorter than the structural relaxation time. At intermediate times—when the monomer displacement is comparable to the bond length—a pronounced peak and then decays slowly as t−1/2, becoming negligible when the displacement is as large as about five bond lengths, i.e. about four monomer diameters or three Kuhn lengths. Local-geometry anisotropy, as measured by the correlation between the initial orientation of a characteristic axis of the Voronoi cell and the subsequent monomer dynamics, is affected at shorter times than the structural relaxation time by the cage shape with antagonistic disturbance by the connectivity. Differently, at longer times, the connectivity favours the persistence of the local-geometry anisotropy, which vanishes when the monomer displacement exceeds the bond length. Our results strongly suggest that the sole consideration of the local order is not enough to understand the microscopic origin of the rattling amplitude of the trapped monomer in the cage of the neighbours.

Pili of Lactobacillus rhamnosus GG mediate interaction with β-lactoglobulin

Recently dairy products like cheese, yogurt or milk proteins are increasingly used to encapsulate probiotic bacteria to promote their survival. The knowledge of interactions occurring between bacteria and milk matrices is essential for the formulation of efficient new products. In this work, Atomic Force Microscopy was used to study interactions between Lactobacillus rhamnosus GG and isolated whey proteins: β-lactoglobulin (β-LG), α-lactalbumin (α-LA) and bovine serum albumin (BSA). For the first time, it was clearly demonstrated that L. rhamnosus GG is able to interact with the β-LG but not with α-LA and BSA. To identify the bacterial surface biomolecules involved in interaction with β-LG, two of its surface mutants were used: a pili depleted strain (L. rhamnosus GG spaCBA) and an exopolysaccharides (EPS) depleted strain (L. rhamnosus GG welE). Two predictive models, FJC and WLC, were used to describe behaviors of the molecule stretched by providing parameters like contour length (Lc), adhesion force (Fadh), Kuhn length (lk) and Persistence length (lp). The absence of adhesion events between β-LG and mutants having no pili reflects the crucial role of this biomolecule for interactions with the protein.

Proving the Role of Entropic Elasticity in Protein Folding

Force-spectroscopy has been an essential tool for understanding how proteins fold and unfold, and their involvement in regulating cell responses to mechanical stimuli. Hence, it is of significance to establish an accurate free energy model of proteins under force capable of describing the mechanisms of protein folding. Even though numerous models have been proposed, the role of polymer physics and the changes in entropy involved in protein elasticity are still controversial. We have developed a model to construct the free energy of elastic proteins under force based on protein folding and polymer physics. The model is constructed by combining Morse and Gaussian potentials describing the enthalpic interactions in the folded state, and the freely jointed-chain (FJC) energy model accounting for the entropic changes during unfolding and stretching. The free parameters of each component are dependent on the protein and thus regulate its mechanical properties [J. Valle-Orero et al, BBRC 460 (2015)]. Here we aim to prove the accuracy of the model by reproducing force-spectroscopy data of a model protein, protein L (B1 domain from Peptostreptococcus magnus). We experimentally fit the force dependent extension of unfolded domains with the FJC using a contour length of 16.3 nm and a Kuhn length of 1.1 nm. The Morse depth of 2 kBT was established by measuring the probability that a domain folds at a given force. A Gaussian barrier of 10.5 kBT was chosen by comparing the experimental unfolding kinetics to those obtained using Brownian dynamics on our energy model. The excellent agreement of our model with experimental data validates and reinforces the theoretical foundations of our physics model. Furthermore, we demonstrate that changes in entropy during extending or collapsing a protein results in departure from simple Bell-like models.

Flexible polyelectrolyte chain in a strong electrolyte solution: Insight into equilibrium properties and force-extension behavior from mesoscale simulation.

Macromolecules with ionizable groups are ubiquitous in biological and synthetic systems. Due to the complex interaction between chain and electrostatic decorrelation lengths, both equilibrium properties and micro-mechanical response of dilute solutions of polyelectrolytes (PEs) are more complex than their neutral counterparts. In this work, the bead-rod micromechanical description of a chain is used to perform hi-fidelity Brownian dynamics simulation of dilute PE solutions to ascertain the self-similar equilibrium behavior of PE chains with various linear charge densities, scaling of the Kuhn step length (lE) with salt concentration cs and the force-extension behavior of the PE chain. In accord with earlier theoretical predictions, our results indicate that for a chain with n Kuhn segments, lE ∼ cs (-0.5) as linear charge density approaches 1/n. Moreover, the constant force ensemble simulation results accurately predict the initial non-linear force-extension region of PE chain recently measured via single chain experiments. Finally, inspired by Cohen's extraction of Warner's force law from the inverse Langevin force law, a novel numerical scheme is developed to extract a new elastic force law for real chains from our discrete set of force-extension data similar to Padè expansion, which accurately depicts the initial non-linear region where the total Kuhn length is less than the thermal screening length.